2013/10/24 by John Calabrese, Calabrese, John, Michael Groechenig +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1310.6600
openalex publication_date 2013/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statements for categories of sheaves twisted by a Gm-gerbe. Our methods allow us to treat even gerbes not coming from a Brauer class. As a pleasant consequence, we deduce a Morita theory for sheaves of abelian categories.